(*******
Find the equation of the tangent plane and the normal line to the sphere x^2 + y^2 + z^2 = 6 at point (1, 1, 2)
*****)

F = (x^2)+ (y^2)+ (z^2);
grad[x_,y_,z_] = {D[F,x],D[F,y],D[F,z]}

{2 x,2 y,2 z}

p = {1,1,2};
n = grad[1,1,2];
n.{x,y,z} == n.p

(***
Normal line:
**)
r[t_] = p + t n
graph=Plot3D[(Sqrt[6-(x^2)-(y^2)]),{x,0,1.5`},{y,0,1.5`},AxesLabel->{x,y,z}];

{1+2 t,1+2 t,2+4 t}
nline=ParametricPlot3D[r[t],{t,-0.25`,0.25`}];

tplane=Plot3D[(1/2) (6-x-y),{x,0.5`,1.5`},{y,0.5`,2.5`}];
Show[graph,tplane,nline,BoxRatios->Automatic,ViewPoint->{2,-1.5`,1.5`},AxesLabel->{x,y,z}]